TITLE: Does the Mandelbrot fractal contain countably or uncountably many copies of itself?
QUESTION [12 upvotes]: I've been working on a program that draws fractal images, and I was struck by a question that came to mind.
It is clear that the Mandelbrot fractal contains infinitely many copies of itself, but I've been wondering, is it a countable or uncountable infinity?

REPLY [3 votes]: As @Sheldon suggested, I looked at the "Mandelbulb" countable question, and the answer given there is awesome, some I'm adding it here as "Community WIKI".
I like the way the answers above provide so much new information, but I love the simplicity of this explanation.

I don't know if you have a precise definition of "bulb", but it's
  reasonable to expect that any bulb ought to contain a sufficiently
  small ball. Any ball contains a point with rational coordinates,
  because the latter are dense. Assuming bulbs are disjoint, this lets
  us define an injection from bulbs to rational points, which are
  countable, therefore there cannot be uncountably many bulbs.